This invention relates generally to simulating fluid flow in a porous medium and, more specifically, to a flow-based method of scaling-up permeability associated with a fine-grid system representative of the porous medium to permeability associated with a coarse-grid system also representative of the porous medium.
Numerical simulation is widely used in industrial fields as a method of simulating a physical system by using a computer. In most cases, there is a desire to model the transport processes occurring in the physical systems. What is being transported is typically mass, energy, momentum, or some combination thereof. By using numerical simulation, it is possible to reproduce and observe a physical phenomenon and to determine design parameters without actual laboratory experiments or field tests. It can be expected therefore that design time and cost can be reduced considerably.
One type of simulation of great interest is a process of inferring the behavior of a real hydrocarbon-bearing reservoir from the performance of a numerical model of that reservoir. The objective of reservoir simulation is to understand the complex chemical, physical, and fluid flow processes occurring in the reservoir sufficiently well to predict future behavior of the reservoir to maximize hydrocarbon recovery. Reservoir simulation often refers to the hydrodynamics of flow within a reservoir, but in a larger sense reservoir simulation can also refer to the total petroleum system which includes the reservoir, injection wells, production wells, surface flowlines, and surface processing facilities.
The principle of numerical simulation is to numerically solve equations describing a physical phenomenon by a computer. Such equations are generally ordinary differential equations and partial differential equations. These equations are typically solved using numerical methods such as the finite difference method, the finite element method, and the finite volume method among others. In each of these methods, the physical system to be modeled is divided into smaller cells (a set of which is called a grid or mesh), and the state variables continuously changing in each cell are represented by sets of values for each cell. An original differential equation is replaced by a set of algebraic equations to express the fundamental principles of conservation of mass, energy, and/or momentum within each smaller unit or cells and of mass, energy, and/or momentum transfer between cells. These equations can number in the millions. Such replacement of continuously changing values by a finite number of values for each cell is called xe2x80x9cdiscretizationxe2x80x9d. In order to analyze a phenomenon changing in time, it is necessary to calculate physical quantities at discrete intervals of time called timesteps, irrespective of the continuously changing conditions as a function of time. Time-dependent modeling of the transport processes proceeds in a sequence of timesteps.
In a typical simulation of a reservoir, solution of the primary unknowns, typically pressure and phase saturation or composition, are sought at specific points in the domain of interest. Such points are called xe2x80x9cgridnodesxe2x80x9d or more commonly xe2x80x9cnodes.xe2x80x9d Cells are constructed around such nodes, and a grid is defined as a group of such cells. The properties such as porosity and permeability are assumed to be constant inside a cell. Other variables such as pressure and phase saturation are specified at the nodes. A link between two nodes is called a xe2x80x9cconnection.xe2x80x9d Fluid flow between two nodes is typically modeled as flow along the connection between them.
In conventional reservoir simulation, most grid systems are structured. That is, the cells have similar shape and the same number of sides or faces. Most commonly used structured grids are Cartesian or radial in which each cell has four sides in two dimensions or six faces in three dimensions. While structured grids are easy to use, they lack flexibility in adapting to changes in reservoir and well geometry and often can not effectively handle the spatial variation of physical properties of rock and fluids in the reservoir. Flexible grids have been proposed for use in such situations where structured grids are not as effective. A grid is called flexible or unstructured when it is made up of polygons (polyhedra in three dimensions) having shapes, sizes, and number of sides or faces that can vary from place to place. Unstructured grids can conform to complex reservoir features more easily than structured grids and for this reason unstructured grids have been proposed for use in reservoir modeling.
One type of flexible grid that can be used in reservoir modeling is the Voronoi grid. A Voronoi cell is defined as the region of space that is closer to its node than to any other node, and a Voronoi grid is made of such cells. Each cell is associated with a node and a series of neighboring cells. The Voronoi grid is locally orthogonal in a geometrical sense; that is, the cell boundaries are normal to lines joining the nodes on the two sides of each boundary. For this reason, Voronoi grids are also called perpendicular bisection (PEBI) grids. A rectangular grid block (Cartesian grid) is a special case of the Voronoi grid. The PEBI grid has the flexibility to represent widely varying reservoir geometry, because the location of nodes can be chosen freely. PEBI grids are generated by assigning node locations in a given domain and then generating cell boundaries in a way such that each cell contains all the points that are closer to its node location than to any other node location. Since the inter-node connections in a PEBI grid are perpendicularly bisected by the cell boundaries, this simplifies the solution of flow equations significantly. For a more detailed description of PEBI grid generation, see Palagi, C. L. and Aziz, K.: xe2x80x9cUse of Voronoi Grid in Reservoir Simulation,xe2x80x9d paper SPE 22889 presented at the 66th Annual Technical Conference and Exhibition, Dallas, Tex. (Oct. 6-9, 1991).
The mesh formed by connecting adjacent nodes of PEBI cells is commonly called a Delaunay mesh if formed by triangles only. In a two-dimensional Delaunay mesh, the reservoir is divided into triangles with the gridnodes at the vertices of the triangles such that the triangles fill the reservoir. Such triangulation is Delaunay when a circle passing through the vertices of a triangle (the circumcenter) does not contain any other node inside it. In three-dimensions, the reservoir region is decomposed into tetrahedra such that the reservoir volume is completely filled. Such a triangulation is a Delaunay mesh when a sphere passing through the vertices of the tetrahedron (the circumsphere) does not contain any other node. Delaunay triangulation techniques are well known; see for example U.S. Pat. No. 5,886,702 to Migdal et al.
Through advanced reservoir characterization techniques, it is common to model the geologic structure and stratigraphy of a reservoir with millions of grid cells, each populated with a reservoir property that includes, but is not limited to, rock type, porosity, permeability, initial interstitial fluid saturation, and relative permeability and capillary pressure functions. However, reservoir simulations are typically performed with far fewer grid cells. The direct use of fine-grid models for reservoir simulation is not generally feasible because their fine level of detail places prohibitive demands on computational resources. Therefore, a method is needed to transform or to scale up the fine-grid geologic reservoir model to a coarse-grid simulation model while preserving, as much as possible, the fluid flow characteristics of the fine-grid model.
One key fluid flow property for reservoir simulation is permeability. Permeability is the ability of a rock to transmit fluids through interconnected pores in the rock. It can vary substantially within a hydrocarbon-bearing reservoir. Typically, permeabilities are generated for fine-scale models (geologic models) using data from well core samples. For simulation cells, the heterogeneities of the geologic model are accounted for by determining an effective permeability. An effective permeability of a heterogeneous medium is typically defined as the permeability of an equivalent homogeneous medium that, for the same boundary conditions, would give the same flux (amount of fluid flow across a given area per unit time). Determining an effective permeability, commonly called permeability upscaling, is not straightforward. The main difficulty lies in the interdependent influences of permeability heterogeneities in the reservoir and the applied boundary conditions.
Many different upscaling techniques have been proposed. Most of these techniques can be characterized as (1) direct methods or (2) flow-based methods. Examples of direct methods are simple averaging of various kinds (e.g., arithmetic, geometric and harmonic averaging) and successive renormalization. The flow-based techniques involve the solution of flow equations and account for spatial distribution of permeability. In general, the flow-based methods require more computational effort but are more accurate than the direct methods.
An overview of different upscaling techniques is provided in the following papers: Wen, X. H. and Gomez-Hernandez, J. J., xe2x80x9cUpscaling Hydraulic Conductivities in Heterogeneous Media: An Overview,xe2x80x9d Journal of Hydrology, Vol. 183 (1996) 9-32; Begg, S. H.; Carter, R. R. and Dranfield, P., xe2x80x9cAssigning Effective Values to Simulator Gridblock Parameters for Heterogeneous Reservoirs,xe2x80x9d SPE Reservoir Engineering (November 1989) 455-465; Durlofsky, L. J., Behrens, R. A., Jones, R. C., and Bernath, A., xe2x80x9cScale Up of Heterogeneous Three Dimensional Reservoir Descriptions,xe2x80x9d Paper SPE 30709 presented at the Annual Technical Conference and Exhibition, Dallas, Tex. (Oct. 22-25, 1995); and Li, D., Cullick, A., Lake, L. W., xe2x80x9cGlobal Scale-up of Reservoir Model Permeability with Local Grid Refinementxe2x80x9d, Journal of Petroleum Science and Engineering, Vol. 14 (1995) 1-13. The upscaling techniques proposed in the past were primarily focused on structured grids. A need exists for a method of upscaling permeabilities associated with a fine-scale geologic model to permeabilities associated with an unstructured, coarse-scale reservoir simulation model.
A method is provided for scaling up permeabilities associated with a fine-scale grid of cells representative of a porous medium to permeabilities associated with an unstructured coarse-scale grid of cells representative of the porous medium. The first step is to generate an areally unstructured, Voronoi, computational grid using the coarse-scale grid as the genesis of the computational grid. The cells of the computational grid are smaller than the cells of the coarse-scale grid and each cell of the computational grid and the coarse-scale grid has a node. The computational grid is then populated with permeabilities associated with the fine-scale grid. Flow equations, preferably single-phase, steady-state pressure equations, are developed for the computational grid, the flow equations are solved, and inter-node fluxes and pressure gradients are then computed for the computational grid. These inter-node fluxes and pressure gradients are used to calculate inter-node average fluxes and average pressure gradients associated with the coarse-scale grid. The inter-node average fluxes and average pressure gradients associated with the coarse grid are then used to calculate upscaled permeabilities associated with the coarse-scale grid.
In a preferred embodiment, the computational grid is constructed from the coarse-scale grid to produce inter-node connections of the computational grid that are parallel to the inter-node connections of the coarse-scale grid. The cells of the computational grid are preferably approximately the same size as the fine-scale cells. The computational grid is preferably populated with permeabilities by assigning to a given node of the computational grid, a predetermined permeability of a cell of the fine-scale grid that would contain the given node""s location if the computational grid were superimposed on the fine-scale grid. The flow equations that are developed for the computational grid are preferably single-phase, steady-state equations. The inter-node average fluxes and average pressure gradients associated with the coarse-scale grid are preferably calculated using only the inter-node connections of the computational grid that fall within a predetermined sub-domain of the computational grid, and more preferably such calculations are made using only the inter-node connections of the computational grid that are parallel to the inter-node connection of the coarse-scale grid associated with the subdomain. The permeabilities associated with inter-node connections of the coarse-scale grid are preferably determined by calculating the ratio of the inter-node average fluxes to the inter-node average pressure gradients that were calculated for the coarse-scale grid.